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Communications in Mathematical Physics

, Volume 208, Issue 3, pp 761–770 | Cite as

Second Eigenvalue of Schrödinger Operators¶and Mean Curvature

  • Ahmad  El Soufi
  • Saïd Ilias

Abstract:

Let $M$ be a compact immersed submanifold of the Euclidean space, the hyperbolic space or the standard sphere. For any continuous potential q on M, we give a sharp upper bound for the second eigenvalue of the operator −Δ+q in terms of the total mean curvature of M and the mean value of q. Moreover, we analyze the case where this bound is achieved. As a consequence of this result we obtain an alternative proof for the Alikakos–Fusco conjecture concerning the stability of the interface in the Allen–Cahn reaction diffusion model.

Keywords

Euclidean Space Diffusion Model Hyperbolic Space Alternative Proof Reaction Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Ahmad  El Soufi
    • 1
  • Saïd Ilias
    • 1
  1. 1.Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France. E-mail: elsoufi@univ-tours.fr; ilias@univ-tours.frFR

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